This application relates to digital communication. More particularly, this application relates to blind equalization of quadrature amplitude modulated signals with an unknown constellation.
Communication involves a transmitter encoding a message in a signal which is sent across a channel to a receiver. Depending on the channel characteristics, the signal may be corrupted when traveling through the channel. A key challenge in communication theory is how to overcome this corruption, to reliably and efficiently extract the message from a received signal. An equalizer is a device designed to compensate for signal corruption by tuning a set of filter values used to filter the received signal. In some applications, the equalizer tunes its filter values using a training signal (e.g., a signal sent before the message signal and whose uncorrupted version is known by the equalizer). In applications where the transmitter does not send a training signal, the equalization is referred to as “blind equalization.”
The process by which the equalizer tunes its filter values may depend on how the transmitter modulates an electronic signal to encode the information in the message. Electronic signals used in communication are generally characterized by at least three well-known properties—frequency, amplitude, and phase—any one of which can be used to represent information. For example, AM radio stations use electromagnetic signals that contain information in their amplitudes, and FM radio stations use electromagnetic signals that contain information in their frequencies. In general, the process of producing a signal that contains information in its frequency, amplitude, and/or phase is called modulation, and the counterpart process of retrieving information from such a signal is called demodulation or detection. For example, for a radio station, AM stands for amplitude modulation and FM stands for frequency modulation.
AM and FM communication protocols are mainly used in radios that communicate analog audio information, and other communication protocols are generally used to communicate digital information. One popular digital communication protocol is known as quadrature amplitude modulation (QAM), in which the communication signal is a combination of two amplitude-modulated sinusoidal signals that have the same frequency but that are π/2 radians apart in phase, i.e., “in quadrature.” Another name for QAM is I/Q modulation, where the “I” refers to the sinusoidal signal “in phase” and the “Q” refers to the sinusoidal signal “in quadrature” with respect to the in-phase signal.
Electronic signals in communication, including QAM signals, can be represented as complex functions, i.e., functions that have both a real and an imaginary part. Complex functions can be plotted on a complex plane in which the horizontal axis represents the real part of the function and the vertical axis represents the complex part of the function. The use of complex functions is a conceptual tool that provides a convenient way to represent signal amplitude and phase. For example, suppose a signal is characterized by the complex function s(t)=r(t)+j·m(t), where r(t) and m(t) are real-valued functions and j designates the imaginary part of the complex function. Using the complex function, the magnitude of the signal s(t) can be computed by |s(t)=√{square root over ((r(t))2+(m(t))2)}{square root over ((r(t))2+(m(t))2)}, and the phase of the signal can be computed by
      ∠    ⁢                  ⁢          s      ⁡              (        t        )              =            arctan      ⁡              (                              m            ⁡                          (              t              )                                            r            ⁡                          (              t              )                                      )              .  As another example, operations that change the frequency content of a signal can also be described using complex numbers and Fourier transforms. For example, one skilled in the art will recognize that multiplying a signal s(t) with the complex sinusoid ejωct=cos(ωct)+j·sin(ωct) in the time domain will produce a resulting signal s′(t)=s(t)ejωct, in which the frequency content of s′(t) is shifted by ωc compared to s(t).
In QAM, the two signals transmitted in quadrature may be cosine and sine signals, each having a particular amplitude. The communication signal, which is a combination of these two component signals, may be expressed theoretically ass(t)=sl,kδ(t−kT)cos(ωct)+sQ,kδ(t−kT)sin(ωct),where ωc is the angular frequency,
  T  =            2      ⁢                          ⁢      π              ω      c      is the sampling interval, k is an integer sample index, δ(t−kT) is the time-shifted ideal impulse function, and sl,k and sQ,k are real values representing the amplitudes of, respectively, the cosine and sine signals at sample k.
Each available pair of values (sl,k, sQ,k) is called a “signal point,” and the set of all available signal points is called a “constellation.” Constellations are commonly represented by plotting the available signal points on a two-dimensional graph with the horizontal axis representing possible values of sl,k and the vertical axis representing possible values of sQ,k. The number M of signal points in a constellation determines the amount of information that is associated with each signal point. In general, each signal point can represent b=└ log2 M┘ bits of information, where M is usually a power of two. If the number of bits per signal point is even, for example when M is 16, 64, or 256, transmitters often use “square constellations”—constellations whose signal points form a square when graphed. If the number of bits is odd, for example when M is 32 or 128, transmitters often use “cross constellations”—constellations whose signal points form a cross when graphed, e.g., a square without points at its corners.
Equalization of QAM signals usually requires the receiver to know which specific constellation the transmitter used to modulate the signal. Using this knowledge, the receiver performs an equalization training algorithm, which implements a least mean square (LMS) algorithm, designed to minimize mean squared detection error, to tune the filter values of its equalizer. In particular, the LMS algorithm can be a function of a constant R, whose value varies with the constellation size.
In certain applications, the receiver not only contends with signal corruption, but also does not know which constellation the transmitter used to perform QAM. Instead, the receiver knows only that the constellation is one of a set of different square and cross constellations, and typically equalizes the channel by assuming in turn which constellation is correct and applying the training algorithm with the appropriate R value corresponding to each assumption. When the receiver assumes an incorrect constellation, the set of filter values typically does not converge to a set of values during the training algorithm and the signal-to-noise ration (SNR) remains relatively low. Only when the receiver assumes the correct constellation do the filter values converge during the training algorithm to equalize the channel and yield a relatively high SNR.
Cable television (TV) is an example of an application requiring blind equalization with an unknown constellation. A cable TV box may receive signals from multiple providers, each of whom may use a different constellation to modulate a transmitted signal. Each time a user changes channels when watching cable TV, the receiver may equalize and decode the newly received signal, without a training signal and without knowing the underlying constellation, to create the visual image seen by the user on his TV. Because the equalization process must finish before the user can view the visual image, the amount of time required for equalization can be an important performance metric for any equalizer design.
The current equalization process requires the receiver to perform the training algorithm up to as many times as the number of possible constellations, requiring a relatively significant number of computations. As communication standards evolve, additional constellations may be used, further increasing the number of possible constellations and thus the number of computations necessary to perform blind equalization. A need remains for a more efficient method of performing blind equalization of QAM signals with an unknown constellation.